Longitudinal and bending oscillations of a three-layered plate with compressible filler contacting with a viscous liquid layer

Deformable body mechanics


Grushenkova E. D.*, Mogilevich L. I.**, Popov V. S.***, Popova A. A.****

Yuri Gagarin State Technical University of Saratov, 77, Politechnicheskaya str., Saratov, 410054, Russia

*e-mail: katenok.09041992@gmail.com
**e-mail: mogilevichli@gmail.com
***e-mail: vic_p@bk.ru
****e-mail: anay_p@bk.ru


The study of a three-layered plate interaction with a layer of pulsating viscous incompressible liquid was performed. The liquid layer motion was regarded as a laminar one occurring in a narrow channel with parallel walls. Sticking conditions at the boundaries of liquid contact with the channel walls were assumed. The upper wall of the channel is being regarded absolutely rigid. The lower wall of the channel represents three-layered plate, and its bearing layers satisfy Kirchhoff's hypotheses. The plate filler is being regarded compressible in the transverse direction. The free support conditions are accepted at the plate end-faces. Oscillations of the three-layered plate are caused by the pressure pulsations at the channel end-faces. Pressure in the liquid layer changes herewith according to harmonic law.

The plane problem of longitudinal and bending hydroelastic oscillations of the three-layered plate was studied. The elastic displacements amplitudes of the three-layer plate were supposed to be much smaller than the thickness of the liquid layer in the channel. On the other hand, the longitudinal size of the channel was supposed to be much bigger compared to its transverse size. The hydroelastic problem consists of dynamics equations of the three-layered plate with compressible filler and liquid dynamics ones. The hydroelasticity problem was formulated. It consists of dynamic equations of the three-layered plate with compressible filler, as well as viscous liquid layer dynamic equations, namely Navier-Stokes and continuity equations. Accounting of normal and shear stresses, acting from the liquid side on the plate bearing layer contacting with it, is being performed while the experiment. Linearization of hydrodynamic equations was performed by the perturbation technique, and solution of the above said problem was obtained for the case of steady-state harmonic oscillations. Hydrodynamic parameters of the liquid layer were determined. Frequency dependent distribution functions of elastic displacements of the plate's layers and pressure of the viscous liquid layer were plotted.


hydroelasticity, oscillations, three-layered plate, compressible filler, viscous liquid, pressure pulsation


  1. Lamb H. On the vibrations of an elastic plate in contact with water, Proceedings of the Royal Society London A, 1920, vol. 98, pp. 205 - 216. DOI: 10.1098/rspa.1920.0064.

  2. Amabili M., Kwak M.K. Free vibrations of circular plates coupled with liquids: revising the Lamb problem, Journal of Fluids and Structures, 1996, vol. 10 (7), pp. 743 - 761. DOI: 10.1006/jfls.1996.0051.

  3. Amabili M. Vibrations of Circular Plates Resting on a Sloshing Liquid: Solution of the Fully Coupled Problem, Journal of Sound and Vibration, 2001, vol. 245 (2), pp. 261 - 283. DOI:10.1006/jsvi.2000.3560.

  4. Askari E., Jeong K.-H., Amabili M. Hydroelastic vibration of circular plates immersed in a liquid-filled container with free surface, Journal of Sound and Vibration, 2013, vol. 332 (12), pp. 3064 - 3085. DOI: 10.1016/j.jsv.2013.01.007.

  5. Alekseev V.V., Indeitsev D.A., Mochalova Yu.A. Zhurnal tekhnicheskoi fiziki, 2002, vol. 72, no. 5, pp. 16 - 21.

  6. Ankilov A.V., Vel'misov P.A., Tamarova Yu.A. Zhurnal Srednevolzhskogo matematicheskogo obshchestva, 2016, vol. 18, no. 1, pp. 94 - 107.

  7. Bochkarev S.A., Lekomtsev S.V., Matveenko V.P. Izvestiya RAN. Mekhanika zhidkosti i gaza, 2016, no. 6, pp. 108 - 120. DOI: 10.7868/S0568528116060049.

  8. Avramov K.V., Strel'nikova E.A. Prikladnaya mekhanika, 2014, vol. 50, no. 3, pp. 86 - 93.

  9. Haddara M.R., Cao S.A. Study of the Dynamic Response of Submerged Rectangular Flat Plates, Marine Structures, 1996, vol. 9 (10), pp. 913 - 933. DOI: 10.1016/0951-8339(96)00006-8.

  10. Chapman C.J., Sorokin S.V. The forced vibration of an elastic plate under significant fluid loading, Journal of Sound and Vibration, 2005, vol. 281 (3), pp. 719 - 741. DOI:10.1016/j.jsv.2004.02.013.

  11. Ergin A. Ugurlu B. Linear vibration analysis of cantilever plates partially submerged in fluid, Journal of Fluids and Structures, 2003, vol. 17, pp. 927 - 939. DOI:10.1016/S0889-9746(03)00050-1.

  12. Dobryanskii V.N., Rabinskii L.N., Radchenko V.P., Solyaev Yu.O. Trudy MAI, 2018, no. 101, available at: http://trudymai.ru/eng/published.php?ID=98252

  13. Kozlovsky Y. Vibration of plates in contact with viscous fluid: Extension of Lamb's model, Journal of Sound and Vibration, 2009, vol. 326, pp. 332 - 339. DOI: 10.1016/j.jsv.2009.04.031.

  14. Önsay T. Effects of layer thickness on the vibration response of a plate-fluid layer system, Journal of Sound and Vibration, 1993, vol. 163, pp. 231 - 259. DOI: 10.1006/jsvi.1993.1162.

  15. Faria C.T., Inman D. J. Modeling energy transport in a cantilevered Euler-Bernoulli beam actively vibrating in Newtonian fluid, Mechanical Systems and Signal Processing, 2014, vol. 45, pp. 317 - 329. DOI: 10.1016/j.ymssp.2013.12.003.

  16. Mogilevich L.I., Popov V.S. Izvestiya RAN. Mekhanika zhidkosti i gaza, 2011, no. 3, pp. 42 - 55.

  17. Ageev R.V., Mogilevich L.I., Popov V.S., Popova A.A. Trudy MAI, 2014, no. 78, available at: http://trudymai.ru/eng/published.php?ID=53466

  18. Alekseev V.V., Indeitsev D.A., Mochalova Yu.A. Zhurnal tekhnicheskoi fiziki, 1999, vol. 69, no. 8, pp. 37 - 42.

  19. Mogilevich L.I., Popov V.S., Popova A.A., Christoforova A.V. Mathematical Modeling of Hydroelastic Oscillations of the Stamp and the Plate, Resting on Pasternak Foundation, IOP Conf. Series: Journal of Physics: Conf. Series, 2018, vol. 944, 012081. DOI :10.1088/1742-6596/944/1/012081.

  20. Starovoitov E.I., Lokteva N.A., Starovoitova N.A. Trudy MAI, 2014, no. 77, available at: http://trudymai.ru/eng/published.php?ID=53018

  21. Gorshkov A.G., Starovoitov E.I., Yarovaya A.V. Mekhanika sloistykh vyazkouprugoplasticheskikh elementov konstruktsii (Mechanics of layered viscoelastoplastic structural elements), Moscow, Fizmatlit, 2005, 576 p.

  22. Mogilevich L.I., Popov V.S., Starovoitov E.I. Nauka i tekhnika transporta, 2006, no. 2, pp. 56 - 63.

  23. Popov V.S., Mogilevich L.I., Grushenkova E.D. Hydroelastic response of three-layered plate interacting with pulsating viscous liquid layer, Lecture Notes in Mechanical Engineering, 2019, pp. 459 - 467. DOI: 10.1007/978-3-319-95630-5_49.

  24. Chernenko A., Kondratov D., Mogilevich L., Popov V., Popova E. Mathematical modeling of hydroelastic interaction between stamp and three-layered beam resting on Winkler foundation, Studies in Systems, Decision and Control, 2019, vol. 199, pp. 671 - 681. DOI: 10.1007/978-3-030-12072-6_54.

  25. Panovko Ya.G., Gubanova I.I. Ustoichivost' i kolebaniya uprugikh system (Stability and Oscillations of Elastic Systems), Moscow, Nauka, 1987, 352 p.

  26. Loitsyanskii L.G. Mekhanika zhidkosti i gaza (Mechanics of Liquids and Gases), Moscow, Drofa, 2003, 840 p.

  27. Van-Daik M. Metody vozmushchenii v mekhanike zhidkostei (Perturbation methods in fluid mechanics), Moscow, Mir, 1967, 312 p.


mai.ru — informational site MAI

Copyright © 2000-2021 by MAI